Question: Solve for $x$ and $y$ by deriving an expression for $x$ from the second equation, and substituting it back into the first equation. $\begin{align*}5x+9y &= 2 \\ -6x-9y &= -8\end{align*}$
Answer: Begin by moving the $y$ -term in the second equation to the right side of the equation. $-6x = 9y-8$ Divide both sides by $-6$ to isolate $x$ $x = {-\dfrac{3}{2}y + \dfrac{4}{3}}$ Substitute this expression for $x$ in the first equation. $5({-\dfrac{3}{2}y + \dfrac{4}{3}}) + 9y = 2$ $-\dfrac{15}{2}y + \dfrac{20}{3} + 9y = 2$ Simplify by combining terms, then solve for $y$ $\dfrac{3}{2}y + \dfrac{20}{3} = 2$ $\dfrac{3}{2}y = -\dfrac{14}{3}$ $y = -\dfrac{28}{9}$ Substitute $-\dfrac{28}{9}$ for $y$ in the top equation. $5x+9( -\dfrac{28}{9}) = 2$ $5x-28 = 2$ $5x = 30$ $x = 6$ The solution is $\enspace x = 6, \enspace y = -\dfrac{28}{9}$.